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<h1 class="heading"><a href="MATH-2023-OPDE.html"><span class="title">MATH 2023: Ordinary and Partial Differential Equations</span></a></h1>
<p class="byline">Xiaoyi Chen and Wei Zhang</p>
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<li class="link frontmatter"><a href="meta_frontmatter.html" data-scroll="meta_frontmatter" class="internal"><span class="title">Front Matter</span></a></li>
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<a href="ch_first.html" data-scroll="ch_first" class="internal"><span class="codenumber">1</span> <span class="title">Introduction</span></a><ul>
<li><a href="sec_1-intro.html" data-scroll="sec_1-intro" class="internal">Classification of Differential Equations</a></li>
<li><a href="sec_2-intro.html" data-scroll="sec_2-intro" class="internal">Linear and Nonlinear Equation</a></li>
<li><a href="sec_3-intro.html" data-scroll="sec_3-intro" class="internal">Geometrical Aspect</a></li>
<li><a href="sec_4-intro.html" data-scroll="sec_4-intro" class="internal">Motivation</a></li>
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<a href="ch_second.html" data-scroll="ch_second" class="internal"><span class="codenumber">2</span> <span class="title">First Order Ordinary Differential Equations</span></a><ul>
<li><a href="sec2_1.html" data-scroll="sec2_1" class="internal">Linear Equations</a></li>
<li><a href="sec2_2.html" data-scroll="sec2_2" class="internal">Further Discussion of Linear Equations (For reading only)</a></li>
<li><a href="sec2_3.html" data-scroll="sec2_3" class="internal">Separable Equations</a></li>
<li><a href="sec2_4.html" data-scroll="sec2_4" class="internal">Difference Between Linear and Nonlinear Equations</a></li>
<li><a href="sec2_5.html" data-scroll="sec2_5" class="internal">Applications of modeling with first order ODE(For reading only)</a></li>
<li><a href="sec2_6.html" data-scroll="sec2_6" class="internal">Exact Equations and Integrating Factors</a></li>
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<a href="ch_third.html" data-scroll="ch_third" class="internal"><span class="codenumber">3</span> <span class="title">third Order Linear Equations</span></a><ul>
<li><a href="sec3_1.html" data-scroll="sec3_1" class="internal">Homogeneous equations with constant coefficient</a></li>
<li><a href="sec3_2.html" data-scroll="sec3_2" class="internal">Fundamental Solutions of Linear Homogeneous Equations</a></li>
<li><a href="sec3_3.html" data-scroll="sec3_3" class="internal">Linear Independence and Wronskian</a></li>
<li><a href="sec3_4.html" data-scroll="sec3_4" class="internal">Complex roots of the characteristic equations</a></li>
<li><a href="sec3_5.html" data-scroll="sec3_5" class="internal">Repeated Roots: Reduction of Order</a></li>
<li><a href="sec3_6.html" data-scroll="sec3_6" class="internal">Non-homogeneous Equations and Method of Undetermined Coefficients</a></li>
<li><a href="sec3_7.html" data-scroll="sec3_7" class="internal">Variation of Parameters</a></li>
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<a href="ch_four.html" data-scroll="ch_four" class="internal"><span class="codenumber">4</span> <span class="title">Higher Order Linear Equations</span></a><ul>
<li><a href="sec4_1.html" data-scroll="sec4_1" class="internal">General Theory of the <span class="process-math">\(n\)</span>-th Order Linear Equations</a></li>
<li><a href="sec4_2.html" data-scroll="sec4_2" class="internal">Homogeneous Equations with Constant Coefficients</a></li>
<li><a href="sec4_3.html" data-scroll="sec4_3" class="internal">The Method of Undetermined Coefficients</a></li>
<li><a href="sec4_4.html" data-scroll="sec4_4" class="internal">The Method of Variation of Parameters</a></li>
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<a href="ch_five.html" data-scroll="ch_five" class="internal"><span class="codenumber">5</span> <span class="title">Series Solutions of Second Order Linear Equations</span></a><ul>
<li><a href="sec5_1.html" data-scroll="sec5_1" class="internal">Brief Review on Power Series</a></li>
<li><a href="sec5_2.html" data-scroll="sec5_2" class="active">Introduction</a></li>
<li><a href="sec5_3.html" data-scroll="sec5_3" class="internal">Series Solutions Near an Ordinary Point</a></li>
<li><a href="sec5_4.html" data-scroll="sec5_4" class="internal">Euler’s Equation</a></li>
<li><a href="sec5_5.html" data-scroll="sec5_5" class="internal">Series Solution near a Regular Singular Point</a></li>
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<a href="ch_six.html" data-scroll="ch_six" class="internal"><span class="codenumber">6</span> <span class="title">System of First Order Linear Equations</span></a><ul>
<li><a href="sec6_1.html" data-scroll="sec6_1" class="internal">Introduction <span class="process-math">\(\&amp;\)</span> Basic Theory</a></li>
<li><a href="sec6_2.html" data-scroll="sec6_2" class="internal">Homogeneous System with Constant Coefficients</a></li>
<li><a href="sec6_3.html" data-scroll="sec6_3" class="internal">Complex Eigenvalues</a></li>
<li><a href="sec6_4.html" data-scroll="sec6_4" class="internal">Repeated Eigenvalues</a></li>
<li><a href="sec6_5.html" data-scroll="sec6_5" class="internal">Fundamental Matrices</a></li>
<li><a href="sec6_6.html" data-scroll="sec6_6" class="internal">Non-homogeneous linear systems</a></li>
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<a href="ch_seven.html" data-scroll="ch_seven" class="internal"><span class="codenumber">7</span> <span class="title">Partial Differential Equations</span></a><ul>
<li><a href="sec7_1.html" data-scroll="sec7_1" class="internal">Two-Point Boundary Value Problems</a></li>
<li><a href="sec7_2.html" data-scroll="sec7_2" class="internal">Eigenvalue Problems</a></li>
<li><a href="sec7_3.html" data-scroll="sec7_3" class="internal">Fourier Series</a></li>
<li><a href="sec7_4.html" data-scroll="sec7_4" class="internal">The Fourier Convergence Theorem</a></li>
<li><a href="sec7_5.html" data-scroll="sec7_5" class="internal">Even and Odd Functions</a></li>
<li><a href="sec7_6.html" data-scroll="sec7_6" class="internal">Introduction to Partial Differential Equations</a></li>
<li><a href="sec7_7.html" data-scroll="sec7_7" class="internal">1D Heat Equation; Solutions by Separation of Variable and Fourier Series</a></li>
<li><a href="sec7_8.html" data-scroll="sec7_8" class="internal">Other Heat Conduction Problems</a></li>
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<a href="ch_eight.html" data-scroll="ch_eight" class="internal"><span class="codenumber">8</span> <span class="title">Laplace transform</span></a><ul>
<li><a href="sec8_1.html" data-scroll="sec8_1" class="internal">What are Laplace Transforms, and Why?</a></li>
<li><a href="sec8_2.html" data-scroll="sec8_2" class="internal">Finding Laplace Transforms</a></li>
<li><a href="sec8_3.html" data-scroll="sec8_3" class="internal">Finding inverse transforms using partial fractions</a></li>
<li><a href="sec8_4.html" data-scroll="sec8_4" class="internal">Solving ODEs and ODE Systems</a></li>
<li><a href="sec8_5.html" data-scroll="sec8_5" class="internal">Step input and Impulse problems</a></li>
<li><a href="sec8_6.html" data-scroll="sec8_6" class="internal">Laplace transform for PDE (heat equation)</a></li>
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<a href="ch_features.html" data-scroll="ch_features" class="internal"><span class="codenumber">9</span> <span class="title">Examples of PreTeXt features</span></a><ul><li><a href="sec_features-blocks.html" data-scroll="sec_features-blocks" class="internal">Environments and Blocks</a></li></ul>
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<li class="link"><a href="solutions-1.html" data-scroll="solutions-1" class="internal"><span class="codenumber">A</span> <span class="title">Selected Hints</span></a></li>
<li class="link"><a href="solutions-2.html" data-scroll="solutions-2" class="internal"><span class="codenumber">B</span> <span class="title">Selected Solutions</span></a></li>
<li class="link"><a href="appendix-1.html" data-scroll="appendix-1" class="internal"><span class="codenumber">C</span> <span class="title">List of Symbols</span></a></li>
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<main class="main"><div id="content" class="pretext-content"><section class="section" id="sec5_2"><h2 class="heading hide-type">
<span class="type">Section</span> <span class="codenumber">5.2</span> <span class="title">Introduction</span>
</h2>
<p id="p-203">Consider</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
P(x) y^{\prime \prime}+Q(x) y^{\prime}+R(x) y=0.
\end{equation*}
</div>
<p class="continuation">This is variable coefficient second order homogeneous equation. A powerful method to solve it is using power series.</p>
<p id="p-204"><dfn class="terminology">Example</dfn> Find the general solution of</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
y^{\prime}-y=0.
\end{equation*}
</div>
<p class="continuation"><dfn class="terminology">Solution</dfn> <span class="process-math">\(y=C e^{x}\)</span> is the general solution.</p>
<p id="p-205"><dfn class="terminology">Idea:</dfn> Many functions can be expanded as a power series. Thus, we first assume that the solution has a power series expansion:</p>
<div class="displaymath process-math" data-contains-math-knowls="" id="eq5_1">
\begin{equation}
y=\sum_{n=0}^{\infty} a_n x^n.\tag{5.2.1}
\end{equation}
</div>
<p class="continuation">If <span class="process-math">\(a_n\)</span> can be determined, then we have found a solution.</p>
<p id="p-206">Substituting (<a href="" class="xref" data-knowl="./knowl/eq5_1.html" title="Equation 5.2.1">(5.2.1)</a>) into the ODE, we have</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq5_1.html ./knowl/eq5_2.html">
\begin{equation*}
\begin{aligned}
&amp;\sum_{n=0}^{\infty} n a_n x^{n-1}-\sum_{n=0}^{\infty} a_n x^n=0,\\
&amp;\to \sum_{k=-1}^{\infty} (k+1) a_{k+1} x^k-\sum_{n=0}^{\infty} a_n x^n=0,\\
&amp;\to \sum_{n=0}^{\infty} (n+1) a_{n+1} x^n-\sum_{n=0}^{\infty} a_n x^n=0,\\
&amp;\to \sum_{n=0}^{\infty} [(n+1) a_{n+1}-a_n] x^n=0.\\
\end{aligned}
\end{equation*}
</div>
<p class="continuation">If a power series is equal to zero, then all the coefficients must vanish. So we have</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq5_1.html ./knowl/eq5_2.html" id="eq5_2">
\begin{equation}
\begin{aligned}
(n+1) a_{n+1}=a_n \to a_{n+1}=\frac{a_n}{n+1}.
\end{aligned}\tag{5.2.2}
\end{equation}
</div>
<p class="continuation">Equation (<a href="" class="xref" data-knowl="./knowl/eq5_2.html" title="Equation 5.2.2">(5.2.2)</a>) is called the <dfn class="terminology">recurrence relation</dfn>. From this relation, we have</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq5_1.html ./knowl/eq5_2.html">
\begin{equation*}
\begin{aligned}
&amp;n=0: a_1=a_0,\\
&amp;n=1: a_2=\frac{a_1}{2}=\frac{a_0}{1\cdot 2}=\frac{a_0}{2!},\\
&amp;n=2: a_3=\frac{a_2}{3}=\frac{a_0}{3!},\\
&amp;\cdots\\
&amp;a_n=\frac{a_0}{n!},
\end{aligned}
\end{equation*}
</div>
<p class="continuation">where <span class="process-math">\(a_0\)</span> is an arbitrary constant. Thus, the solution is</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq5_1.html ./knowl/eq5_2.html">
\begin{equation*}
y=\sum_{n=0}^{\infty} a_n x^n=a_0 \sum_{n=0}^{\infty} \frac{x^n}{n!}=a_0 e^x.
\end{equation*}
</div>
<p id="p-207">Classification of Types of Points:Consider</p>
<div class="displaymath process-math" data-contains-math-knowls="" id="eq5_3">
\begin{equation}
P(x) y^{\prime \prime}+Q(x) y^{\prime}+R(x) y=0,\tag{5.2.3}
\end{equation}
</div>
<p class="continuation">A point <span class="process-math">\(x_0\)</span> such that <span class="process-math">\(P(x_0) \neq 0\)</span> is called an <dfn class="terminology">ordinary point</dfn>. If <span class="process-math">\(P(x_0)=0\text{,}\)</span> then <span class="process-math">\(x_0\)</span> is a <dfn class="terminology">singular point</dfn>. These are the definitions suitable when <span class="process-math">\(P(x), Q(x)\)</span> and <span class="process-math">\(R(x)\)</span> are polynomials and have no common factors.More generally, the ordinary and singular points are defined as:If at <span class="process-math">\(x=x_0\text{,}\)</span> both <span class="process-math">\(Q(x)/P(x)\)</span> and <span class="process-math">\(R(x)/P(x)\)</span> are analytic, then <span class="process-math">\(x=x_0\)</span> is called an <dfn class="terminology">ordinary point</dfn>. Otherwise, <span class="process-math">\(x=x_0\)</span> is a <dfn class="terminology">singular point</dfn>.</p>
<p id="p-208"><dfn class="terminology">Example 1</dfn>Is <span class="process-math">\(x=0\)</span> an ordinary point or singular point of</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
x y^{\prime \prime}+(\sin x) ~y^{\prime}+x^3 y=0?
\end{equation*}
</div>
<p class="continuation"><dfn class="terminology">Solution:</dfn></p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\begin{aligned}
\frac{Q(x)}{P(x)}&amp;=\frac{\sin x}{x}=\frac{1}{x} \left[x-\frac{x^3}{3!}+\frac{x^5}{5!}+\cdots \right]\\
&amp;=1-\frac{x^2}{3!}+\frac{x^4}{5!}+\cdots,\quad |x|&lt;\infty,\\
\frac{R(x)}{P(x)}&amp;=x^2.
\end{aligned}
\end{equation*}
</div>
<p class="continuation">Since both <span class="process-math">\(Q(x)/P(x)\)</span> and <span class="process-math">\(R(x)/P(x)\)</span> are analytic at <span class="process-math">\(x=0\text{,}\)</span> <span class="process-math">\(x=0\)</span> is an ordinary point.</p>
<p id="p-209"><dfn class="terminology">Example 2</dfn>For</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
(1-x^2) y^{\prime \prime}-2 x y^{\prime}+\alpha (\alpha+1) y=0,
\end{equation*}
</div>
<p class="continuation">where <span class="process-math">\(\alpha\)</span> is a constant. Is <span class="process-math">\(x=1\)</span> an ordinary point or singular point?<dfn class="terminology">Solution:</dfn></p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\frac{Q(x)}{P(x)}=\frac{-2 x}{1-x^2},
\end{equation*}
</div>
<p class="continuation">which is not continuous at <span class="process-math">\(x=1\text{.}\)</span> The Taylor series does not exist at this point, thus <span class="process-math">\(x=1\)</span> is a singular point.</p>
<p id="p-210"><dfn class="terminology">Regular Singular Point and Irregular Singular Point:</dfn>Suppose that <span class="process-math">\(x=x_0\)</span> is a singular point. If both <span class="process-math">\((x-x_0) Q(x)/P(x)\)</span> and <span class="process-math">\((x-x_0)^2 R(x)/P(x)\)</span> are analytic, then <span class="process-math">\(x=x_0\)</span> is called a <dfn class="terminology">regular singular point</dfn>. Otherwise, <span class="process-math">\(x=x_0\)</span> is called an <dfn class="terminology">irregular singular point</dfn>.</p>
<p id="p-211"><dfn class="terminology">Example 3</dfn>For</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
(1-x^2) y^{\prime \prime}-2 x y^{\prime}+\alpha (\alpha+1) y=0,
\end{equation*}
</div>
<p class="continuation">is <span class="process-math">\(x=1\)</span> a regular singular point or irregular singular point?</p>
<p id="p-212"><dfn class="terminology">Solution:</dfn></p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
(x-1) \frac{Q(x)}{P(x)}=(x-1) \frac{-2 x}{1-x^2}=-2 \frac{x}{1+x},
\end{equation*}
</div>
<p class="continuation">which are analytic at <span class="process-math">\(x=1\)</span> because the rational function has no singularities at <span class="process-math">\(x=1\text{.}\)</span> Besides,</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
(x-1)^2 \frac{R(x)}{P(x)}=(x-1)^2 \frac{\alpha (\alpha+1)}{1-x^2}=(1-x) \frac{\alpha (\alpha+1)}{1+x},
\end{equation*}
</div>
<p class="continuation">which is also analytic at <span class="process-math">\(x=1\text{.}\)</span> In conclusion, <span class="process-math">\(x=1\)</span> is a regular singular point.Note: In this course, we will not consider the case of irregular singular points.</p></section></div></main>
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